Approximation algorithms for the graph burning on cactus and directed trees

Abstract

Given a graph [Formula: see text], the problem of Graph Burning is to find a sequence of nodes from [Formula: see text], called a burning sequence, to burn the whole graph. This is a discrete-step process, and at each step, an unburned vertex is selected as an agent to spread fire to its neighbors by marking it as a burnt node. A burnt node spreads the fire to its neighbors at the next consecutive step. The goal is to find the burning sequence of minimum length. The Graph Burning problem is NP-Hard for general graphs and even for binary trees. A few approximation results are known, including a [Formula: see text]-approximation algorithm for general graphs and a [Formula: see text]-approximation algorithm for trees. The Graph Burning on directed graphs is more challenging than on undirected graphs. In this paper, we propose (1) A [Formula: see text]-approximation algorithm for a cactus graph (undirected), (2) A [Formula: see text]-approximation algorithm for multi-rooted directed trees (polytree) and (3) A [Formula: see text]-approximation algorithm for the single-rooted directed tree (arborescence). We implement all three approximation algorithms, and the results are shown for randomly generated cactus graphs, multi-rooted directed trees and single-rooted directed trees.